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Aggregation of evaluations without unanimity

Yuval Filmus

TL;DR

This work extends Arrow-type impossibility results to predicates over arbitrary finite alphabets by unifying Dokow–Holzman and Szegedy–Xu approaches within Mossel's non-unanimity framework. It introduces and analyzes multi-sorted polymorphisms and the notion of $\Phi$-triviality, proving a reduction to the $n=2$ base case and, under a unary-augmentation step, to $n=1$, to classify all polymorphisms of $P$ as either dictatorial or certificate-type. The authors apply these structural results to symmetric binary predicates over $\{0,1\}$, providing a complete classification of their aggregators, including explicit conditions for triviality under different $\Phi$-collections and the existence of AND/OR or Latin square polymorphisms. They further connect $\Phi$-triviality to impossibility domains, clarifying when non-dictatorial unanimity- or non-unanimity-preserving polymorphisms can exist, and offering practical criteria for determining the aggregators of a given symmetric predicate. This framework yields concrete, algorithmic criteria for identifying aggregators and impossibility domains in finite alphabets, with implications for social choice and related areas in universal algebra and CSPs.

Abstract

Dokow and Holzman determined which predicates over $\{0, 1\}$ satisfy an analog of Arrow's theorem: all unanimous aggregators are dictatorial. Szegedy and Xu, extending earlier work of Dokow and Holzman, extended this to predicates over arbitrary finite alphabets. Mossel extended Arrow's theorem in an orthogonal direction, determining all aggregators without the assumption of unanimity. We bring together both threads of research by extending the results of Dokow-Holzman and Szegedy-Xu to the setting of Mossel. As an application, we determine, for each symmetric predicate over $\{0,1\}$, all of its aggregators.

Aggregation of evaluations without unanimity

TL;DR

This work extends Arrow-type impossibility results to predicates over arbitrary finite alphabets by unifying Dokow–Holzman and Szegedy–Xu approaches within Mossel's non-unanimity framework. It introduces and analyzes multi-sorted polymorphisms and the notion of -triviality, proving a reduction to the base case and, under a unary-augmentation step, to , to classify all polymorphisms of as either dictatorial or certificate-type. The authors apply these structural results to symmetric binary predicates over , providing a complete classification of their aggregators, including explicit conditions for triviality under different -collections and the existence of AND/OR or Latin square polymorphisms. They further connect -triviality to impossibility domains, clarifying when non-dictatorial unanimity- or non-unanimity-preserving polymorphisms can exist, and offering practical criteria for determining the aggregators of a given symmetric predicate. This framework yields concrete, algorithmic criteria for identifying aggregators and impossibility domains in finite alphabets, with implications for social choice and related areas in universal algebra and CSPs.

Abstract

Dokow and Holzman determined which predicates over satisfy an analog of Arrow's theorem: all unanimous aggregators are dictatorial. Szegedy and Xu, extending earlier work of Dokow and Holzman, extended this to predicates over arbitrary finite alphabets. Mossel extended Arrow's theorem in an orthogonal direction, determining all aggregators without the assumption of unanimity. We bring together both threads of research by extending the results of Dokow-Holzman and Szegedy-Xu to the setting of Mossel. As an application, we determine, for each symmetric predicate over , all of its aggregators.

Paper Structure

This paper contains 21 sections, 17 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Predicates
  3. Polymorphisms
  4. Triviality
  5. Main result
  6. There exists $(\phi_1,\dots,\phi_m) \in \Phi$ such that $g_i(x) = \phi_i(x_1)$ for all $i$.
  7. There exists $(\phi_1,\dots,\phi_m) \in \Phi$ such that $g_i(x) = \phi_i(x_2)$ for all $i$.
  8. The functions $g_1,\dots,g_m$ conform to some certificate $\rho$.
  9. Reduction to n=1
  10. There exists $(\phi_1,\dots,\phi_m) \in \Phi$ such that $g_i = \phi_i$ for all $i$.
  11. The functions $g_1,\dots,g_m$ conform to some certificate $\rho$.
  12. There exists $(\phi_1,\dots,\phi_m) \in \Phi$ such that $g_i^\eta = \phi_i$ for all $i$.
  13. The functions $g_1^\eta,\dots,g_m^\eta$ conform to some certificate $\rho$.
  14. Symmetric binary predicates: triviality
  15. $P$ is closed under setting some coordinates to constants, and the constant coordinates do not constitute a certificate.
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $f,g,h\colon \{0,1\}^n \to \{0,1\}$ satisfy the following two conditions: Then there is a coordinate $j$ such that $f(x) = g(x) = h(x) = x_j$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2: Mossel
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • ...and 14 more